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Unrefinable Partitions into Distinct Parts and Numerical Semigroups

Lorenzo Campioni

TL;DR

This work studies unrefinable partitions into distinct parts, introducing a bridge to numerical semigroups by matching missing parts with gaps of a semigroup via hookset structures on the associated Young diagram. It develops criteria for recognizing unrefinable partitions using the vector of forbidden elements $\Vec{p}_{\lambda}$ and links the Apéry set $Ap(S,s_1)$ to this vector. It establishes the inclusion $NS \subset \mathcal{U}$ (not conversely), analyzes maximal unrefinable partitions and their decomposition by $\mathrm{mex}$, and shows that when the maximal part $\lambda_t$ is a prime, the corresponding semigroup lies in the symmetric class $SNS$. The results fuse partition theory with numerical semigroup theory, offering a unified framework and inviting further work on generating function approaches and deeper structural connections.

Abstract

This article investigates structural connections between unrefinable partitions into distinct parts and numerical semigroups. By analysing the hooksets of Young diagrams associated with numerical sets, new criteria for recognising unrefinable partitions are established. A correspondence between missing parts and the gaps of numerical semigroups is developed, extending previous classifications and enabling the characterisation of partitions with maximal numbers of missing parts. In particular, the results show that certain families of unrefinable partitions correspond precisely to symmetric numerical semigroups when the maximal part is prime. Further structural consequences, examples, and a decomposition of unrefinable partitions by minimal excludant are discussed, together with implications for the study of maximal unrefinable partitions.

Unrefinable Partitions into Distinct Parts and Numerical Semigroups

TL;DR

This work studies unrefinable partitions into distinct parts, introducing a bridge to numerical semigroups by matching missing parts with gaps of a semigroup via hookset structures on the associated Young diagram. It develops criteria for recognizing unrefinable partitions using the vector of forbidden elements and links the Apéry set to this vector. It establishes the inclusion (not conversely), analyzes maximal unrefinable partitions and their decomposition by , and shows that when the maximal part is a prime, the corresponding semigroup lies in the symmetric class . The results fuse partition theory with numerical semigroup theory, offering a unified framework and inviting further work on generating function approaches and deeper structural connections.

Abstract

This article investigates structural connections between unrefinable partitions into distinct parts and numerical semigroups. By analysing the hooksets of Young diagrams associated with numerical sets, new criteria for recognising unrefinable partitions are established. A correspondence between missing parts and the gaps of numerical semigroups is developed, extending previous classifications and enabling the characterisation of partitions with maximal numbers of missing parts. In particular, the results show that certain families of unrefinable partitions correspond precisely to symmetric numerical semigroups when the maximal part is prime. Further structural consequences, examples, and a decomposition of unrefinable partitions by minimal excludant are discussed, together with implications for the study of maximal unrefinable partitions.
Paper Structure (5 sections, 18 theorems, 24 equations, 1 figure)

This paper contains 5 sections, 18 theorems, 24 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Unrefinable Partitions into Distinct Parts
  4. Numerical Semigroups
  5. Connection between Unrefinable Partitions and Numerical Semigroups

Key Result

Lemma 1

Let $\lambda=(\lambda_1,\ldots,\lambda_t)$ be a partition into distinct parts. If $\#\mathcal{M}_{\lambda}=\{0,1\}$, then $\lambda\in\mathcal{U}$.

Figures (1)

  • Figure 1: Given $\lambda=\{1,2,4,5,7,10,13\}$, any directed path from $\{0\}$ to $\{6,8,9,11,12\}$ gives a sequence of integers that, added to $\lambda$, keep it an unrefinable partition. These are all the unrefinable partitions such that $\lambda_t=13$ and $\mu_1=3$.

Theorems & Definitions (62)

  • Definition 1
  • Definition 2
  • Lemma 1
  • proof
  • Definition 3
  • Corollary 1
  • proof
  • Lemma 2
  • proof
  • Proposition 1
  • ...and 52 more